Math 6620 • Analysis of Numerical Methods II

(20 pts) Consider the PDE,
\[
u_t + au_x = 0,\qquad u(x,0)=u_0(x),\qquad a > 0,
\]
with periodic boundary conditions over the spatial domain \(x \in [0, 1)\). Use the standard
notation, \(u_j^n \approx u(x_j,t^n)\) with \(x_j = jh\) and \(t^n = nk\) with \(h = \frac{1}{M} > 0\) for some \(M \in \mathbb{N}\) and
\(k > 0\). Consider the following semi-Lagrangian scheme:
\[
u_j^{n+1}=p_j(y),\qquad y := X(t^n;x_j,t^{n+1}),
\]
where \(X(t; x, t_0)\) is the time-\(t\) spatial location of the PDE's characteristic curve passing through
spatial location \(x\) at time \(t_0\), and \(p_j(\cdot)\) is the linear interpolant formed by the data \((x_{j-1},u_{j-1}^n)\)
and \((x_j,u_j^n)\).
\begin{enumerate}

\item
(10 pts) Write this scheme as an explicit function of the \(u_j^n\) values.


\item
(10 pts) Use von Neumann stability analysis to determine a stability condition for this
scheme.

\end{enumerate}